Friday, May 9, 2014

Decoherence in a box

Schrodinger considered superposition the characteristic
trait of quantum mechanics:

"When two systems, of which we know the states by their respective representatives, enter into temporary physical interaction due to known forces between them, and when after a time of mutual influence the systems separate again, then they can no longer be described in the same way as before, viz. by endowing each of them with a representative of its own. I would not call that one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought. By the interaction the two representatives [the quantum states] have become entangled.”

So why we don’t see then in our macroscopic world a cat both
dead and alive? In other words, how
does classicality emerge from quantum mechanics? The answer comes (in part) from
decoherence which shows that interaction with environment makes the density matrix diagonal, free of interference terms. But is this a general enough mechanism?

In his talk “A Closed System Perspective for Decoherence”, Sebastian
Fortin presented a framework which generalizes the current approach for both open and closedsystems.

First let’s present the problem with closed systems. Unitary
time evolution in quantum mechanics can prevent canceling the non-diagonal
parts of the density matrix in a closed system. Decoherence works for open systems
by talking advantage of the very large numbers of degrees of freedom of the
environment which can absorb the “unwanted” information. There are two time characteristics
in an open quantum system: decoherence time and relaxation time. You can have decoherence without equilibrium (an
obvious example are the planets of the solar system) but dissipation always
implies decoherence. (This means that a quantum system reaches decoherence faster
than equilibrium.)

But how can we talk about irreversible processes in quantum
mechanics? Doesn’t this contradict unitary time evolution? Non-unitary time
evolution can arise when we sum over (trace over) the degrees of freedom of the
environment. This leads to a (non-unitary) master equation.

In a closed
systems, equilibrium or standard decoherence may not occur, but we can still get emergent classical behavior. So here is Fortin’s proposed solution: non-unitary
time evolution is achieved by course-graining:

split
the information in relevant and irrelevant parts

compute
time evolution of relevant observables

determine if you reach equilibrium (achieve relaxation)

if so, compute
the decay times

This is not at all unusual: for example in the kinetic
theory of gasses individual molecules never rest and reach (static) equilibrium. However,
course graining can extract the relevant macroscopic information: gas density,
pressure, etc.

In a closed quantum system the non-diagonal elements do not go to zero, but they may cancel each
other out.

This mechanism also works in open systems in addition to the
standard way of evolving the non-diagonal elements to zero. Hence, the
framework is universal. As an application: Fortin talked about self-induced
decoherence: